Updates to Chapter 8, "Step Right Up!",
in Dissections: Plane & Fancy, by Greg N. Frederickson

• Fibonacci to Diophantus
• Wrong figure referenced
• Penta-penta need not be integral
• Typos in Method 5
• An earlier dissection
• Cossali's name
• Another way to generate Cossali's class
• Another class of 3 squares to 1
• n Squares to one

Fibonacci to Diophantus

Katharina Huber pointed out that the roles of x and y are reversed when reducing Fibonacci's formula to Diophantus's formula on page 73.

Wrong figure referenced

Koji Miyazaki, Hirohisa Hioki, and Naoki Odaka pointed out a problem in the reference in the fourth and fifth lines from the bottom of page 75. It should state that Figure 8.4 closely resembles Figure 7.10, not Figure 7.5.

Penta-penta need not be integral

As suggested by Puzzle 8.3, the Penta class can include identities produced using nonintegral values for p and q. This observation also applies to the Penta-penta class.

Typos in Method 5

There are two typos in the specification of Method 5. In the second line of step 2, the first expression should be y - q - 1 rather than z - q - 1 and the second expression should be y - x rather than z - x.
(These are corrected in the paperback edition.)

An earlier dissection

Using Method 2C, I gave 5-piece dissections for all squares in the PP-plus class. One solution in the PP-plus class is 9² + 2² + 6² = 11², for which L. P. Mochalov gave a 5-piece dissection in Vladimir Belov's 1992 book. (See the updates to chapter 1.) Mochalov's method can be generalized to handle all squares in the PP-plus class.

Cossali's name

I should have written Cossali's name as Pietro Cossali.

Another way to generate Cossali's class

There is another way to produce Cossali's class without resorting to the device of n=m or n=p, as I did in the book. Instead, take p, q, and n all to be 1, and take m to to be odd. This will give values that have a common factor of 4. The values are then in a different order from what you see on page 84.

Another class of 3 squares to 1

David Eppstein (at the University of California, Irvine) has pointed out that Robert Reid's 5-piece dissection of 4-square, 9-square, and 48-square to a 49-square in Fig. 8.19 is not an isolated dissection. Instead, it belongs to a class whose first four members are (1, 2, 2, 3), (4, 9, 48, 49), (21, 50, 1470, 1471), and (120, 289, 48960, 48961).

Members of the class are integral solutions to the simultaneous equations:

x (z + 1 - y - x) = (x - 1) z
z + 1 - y = (y - 1) (y - x)
x² + y² = 2 z + 1
w = z + 1

All solutions to these equations can be found by the following method. Consider a series of pairs of values a_i,b_i satisfying:

a₁ = b₁ = 1
a_i+1 = a_i + b_i
b_i+1 = 2a_i + b_i

We can prove by induction that:

(b_2j)² = 2(a_2j)² + 1
(b_2j+1)² + 1 = 2(a_2j+1)²

Let y_2j = (b_2j)² and y_2j+1 = (b_2j+1)² + 1. Then take x_i = sqrt(y_i-1(y_i-1)).
It follows that z_i = x_i sqrt(2y_i(y_i-1)).

Thus the values of x_i, y_i, z_i, and w_i satisfy the requirements for Reid's dissection.

Note that b_i / a_i is an approximation for the square root of 2, which is generated by the continued fraction technique for solving the Pell equation b² + 1 = 2a². About 130 A.D. Theon of Smyrna gave the recursive formulas for a_i and b_i. They had earlier been given in geometric form. (See page 341 of Dickson (1920).) Thus I will call this class of solutions to x² + y² + z² = w² Theon's class.

Since the 5-piece dissection for this class uses two steps, and the steps are specializations of P-slides, the existence of this class suggests a relationship for three squares to one that I missed in chapter 4. Squares for that relationship will have 7-piece dissections.

n Squares to one

I identified many dissections of three squares to one that use just five pieces, and also gave one dissection of four squares that uses six pieces. In 1978, A.J.W. Duivestijn published a set of 21 squares of different sizes that can be rearranged without cutting to form a single square. This leaves open the problem of dissecting n different-sized squares, for n between 3 and 21, into the fewest number of pieces that form a larger square. Another optimization criterion is to minimize the total length of the cut. Joe Kingston and Des MacHale investigate this latter problem in their article, "Dissecting Squares," which is to appear in the Mathematical Gazette in November 2001. Since submitting their article, they have found some improved results, which they describe in "Some Improved Dissections of Squares."

Copyright 1997-2003, Greg N. Frederickson.
Permission is granted to any purchaser of Dissections: Plane & Fancy to print out a copy of this page for his or her own personal use.

Last updated October 4, 2003.